MAE101 FINAL EXAM QUESTIONS AND CORRECT ANSWERS

MAE101 FINAL EXAM QUESTIONS AND
CORRECT ANSWERS
Let P(1,2,1), Q(1,0,-1), R(2,2,0) be the vertices of a parallelogram with adjacent sides RP and RQ.
Find the other vertex S. - CORRECT ANSWERS a. None of the other choices is correct



Which of the vectors below is orthogonal to both (2, 1, -1) and (-3, -2, 4)? - CORRECT
ANSWERS d. (2, -5, -1)



Find the shortest distance between the pair of nonparallel lines [x y z]T = [1 -1 -1]T+t[2 3 1]T
and [x y z]T = [1 -2 -1]T+t[3 2 2]T . - CORRECT ANSWERS 1 / sqrt(42)



Find the area of the triangle with the following vertices A(1, 1, -1), B(2, 0, 1), C(1, -1, 3). -
CORRECT ANSWERS sqrt 5



Let u = (1, 2, 1); v = (2, 3, 0); w = (0, 1, 0). Find projection of u x v on w. - CORRECT
ANSWERS a. (0, 2, 0)



The volume of the pyramid with vertices (0; 0; 0); (-2; 8; 14); (-6; 7;-3) and (4; 0; 2) is: - CORRECT
ANSWERS b. 70



Let be projection on the line y = 2x followed by reflection in the x-axis and let the vector v = [1
1]T. Find T(v). - CORRECT ANSWERS 3/5 -6/5



Let u = (3, 3, 6), v = (4, 4, 3), w = (-6, 3, 3) and x = (41, 5, 12). We can writex = au + bv + cw,where
a, b, c are numbers. Find b. - CORRECT ANSWERS d. 2

,Which of the following are subspaces of R3?(i) {(x, y, z)| 2x-y+3z=0}(ii) (x, y, z)| xy=0} - CORRECT
ANSWERS The correct answer is: (i) only



Find the dimension of the subspace U = {[x, y, z, t] | x+4y-z = 0; x-2z+ t= 0}. - CORRECT
ANSWERS The correct answer is: 2



Find the dimension of the subspace

H = { [a+2b+2d, c+d, -3a-6b+4c-2d, -c-d] | a, b, c, d in R} - CORRECT ANSWERS The correct
answer is: dim H = 2



Let the set of vectors {u, v, w} in R3 be independent. Which of the followings are true?

(i) {u; v; u-v+w} is also independent.

(ii) {u; v+w; u+v+w} is also independent. - CORRECT ANSWERS The correct answer is: (i)
ony



Let X = [a, b, c]. Let U be the subspace spanned by the orthogonal basis{u = [1, 1, 1], v = [1, -1,
0], w = [1, 1, -2]}.Find the coefficient of u when expressing X as a linear combination of {u, v, w}.
- CORRECT ANSWERS The correct answer is: (a+b+c)/3



Find the dimension of the null space of the matrix

[1 -2 3 -3 -1 / -2 5 -5 4 -4 /-1 3 -2 1 -5 ] - CORRECT ANSWERS The correct answer is: 3



Let A be 4 x 7 matrix that has dim(Null(A)) = 5. Choose the correct statements:(i) A has exactly 5
independent columns(ii) A has exactly 2 independent rows - CORRECT ANSWERS The
correct answer is: (ii)



Which of the following points lie on the line with parametric equations

x = 2+4t; y = 3-7t; z = 5t?

, P(10; -11; 10); Q(-2; 4; 5); R(-2; 10; -5) - CORRECT ANSWERS The correct answer is: P and
R



Find an equation of the plane which contains the point (2, 4, 3) and which is perpendicular to
the planes x+2y-z=1, 3x-4y = 2 - CORRECT ANSWERS The correct answer is: 4x+3y+10z=50



Find the shortest distance between the pair of parallel lines [x y z]T = [1 1 1]T+t[3 0 4]T and [x y
z]T = [0 1 0]T+t[3 0 4]T - CORRECT ANSWERS The correct answer is: 1/5



Find the area of the triangle with vertices A(1, 2, 1); B(3; 2, 1), C(0, 5, 2). - CORRECT
ANSWERS sqrt 10



Let u = (1; 1; 1); v = (0; 1; 1) and w = (1; 0; 1). Find the length of x = (3u + v) w. - CORRECT
ANSWERS sqrt 33



Consider the points A(1; 2; 3); B(1; 3; 2) and C(2; 1; 3). Find a point D on the Z-axis so that

the volume of the tetrahedron ABCD is 3. - CORRECT ANSWERS The correct answer is: (0,
0 -12)



If : T: R2 -> R2 T[x y ] = [ x- y/2 y-x/2]

Determine if T is projection on a line, reflection in a line, or rotation through an angle, and find
the line or angle; - CORRECT ANSWERS The correct answer is: Projection on the line y = -x



Determine whether x = (5, 6) lies in U= span{u=(1, 2); v=(0, 1); w=(2, 3)}. If so, write x = a.u+ b.v
+c.w then find a+b+c. - CORRECT ANSWERS The correct answer is: a+b+c = 1



Which of the following are subspaces of R2

(i) {(x,y): 2x+7y = 0}